Stationary axisymmetric exteriors for perturbations of isolated bodies in general relativity, to second order

TL;DR

This paper develops second-order perturbation theory for stationary axisymmetric bodies in general relativity, deriving boundary conditions and compatibility criteria for matching interior and exterior solutions.

Contribution

It introduces second-order perturbation analysis for axisymmetric bodies with detailed boundary conditions and compatibility criteria in the context of general relativity.

Findings

Derived perturbation of matching conditions as boundary conditions

Established necessary and sufficient conditions for boundary compatibility

Provided detailed analysis for spherical body perturbations

Abstract

Perturbed stationary axisymmetric isolated bodies, e.g. stars, represented by a matter-filled interior and an asymptotically flat vacuum exterior joined at a surface where the Darmois matching conditions are satisfied, are considered. The initial state is assumed to be static. The perturbations of the matching conditions are derived and used as boundary conditions for the perturbed Ernst equations in the exterior region. The perturbations are calculated to second order. The boundary conditions are overdetermined: necessary and sufficient conditions for their compatibility are derived. The special case of perturbations of spherical bodies is given in detail.